Question

Is { [a a+b a+b b] | a,b belongs to Z} a subring of M2(Z)? Why, or why not?

Answer 1

$\begin{pmatrix}a&a+b\\a+b&b\end{pmatrix} \begin{pmatrix}m&m+n\\m+n&n\end{pmatrix} =$

$=\begin{pmatrix}am+(a+b)(m+n)&a(m+n)+(a+b)n\\m(a+b)+b(m+n)&(m+n)(a+b)+bn\end{pmatrix}$

Off-diagonal elements

$a(m+n)+(a+b)n \ne m(a+b)+b(m+n)$

Hence, it is not a subring because it is not closed under multiplication.